Gravitational lensing by self-dual black holes in loop quantum gravity

We study gravitational lensing by a recently proposed black hole solution in loop quantum gravity. We highlight the fact that the quantum gravity corrections to the Schwarzschild metric in this model evade the “mass suppression” effects (that the usual quantum gravity corrections are susceptible to) by virtue of one of the parameters in the model being dimensionless, which is unlike any other quantum gravity motivated parameter. Gravitational lensing in the strong and weak deflection regimes is studied, and a sample consistency relation is presented which could serve as a test of this model. We discuss that, though the consistency relation for this model is qualitatively similar to what would have been in Brans–Dicke, in general it can be a good discriminator between many alternative theories. Although the observational prospects do not seem to be very optimistic even for a galactic supermassive black hole case, time delay between relativistic images for a billion solar mass black holes in other galaxies might be within reach of future relativistic lensing observations.

Figure 1

The radius of photon sphere (scaled by Schwarzschild radius) as a function of $\epsilon$ characterizing quantum gravity corrections.

Figure 2

The strong lensing coefficients as a function of $\epsilon$ characterizing quantum gravity corrections.

Figure 3

The strong lensing observables as a function of $\epsilon$ characterizing quantum gravity correction for the galactic supermassive black hole scenario. $r$ is in magnitude, and ${\theta }_{\infty }$ and $s$ are in microarcseconds.

Figure 4

The time delay (in dimensionless units) between the first and second relativistic Einstein rings as a function of $\epsilon$ characterizing quantum gravity correction